Four-manifolds with positive isotropic curvature

Abstract

In [Bre19], Simon Brendle showed that any compact manifold of dimension n ≥ 12 with positive isotropic curvature and contains no nontrivial incompressible (n − 1)− dimensional space form is diffeomorphic to a connected sum of finitely many spaces, each of which is a quotient of S n or S n−1 × R by standard isometries. We show that this result is actually true for n ≥ 9.

Classifying sufficiently connected PSC manifolds in 4 and 5 dimensions

“…In higher dimensions, Brendle and Schoen [5] and Nguyen [27] proved the PIC condition is preserved under the Ricci flow; this is an important ingredient in Brendle and Schoen's proof of the differentiable sphere theorem. Recently, Brendle has achieved a breakthrough in the study of the Ricci flow of PIC manifolds and has extended Hamilton's result to dimensions n 12 [4]; as above, this result has been used to prove the Gromov-Schoen conjectures for n 12 by Huang [20].…”

Vacuum Static Spaces with Positive Isotropic Curvature

“…On the other hand, PIC implies that g has positive scalar curvature [25]. Recently, there are many known results on the structure of Riemannian manifolds with positive isotropic curvature [4], [6], [7], [12], [13], [28], [29].…”

Besse conjecture with positive isotropic curvature

“…On the contrary, a PIC implies that g has a positive scalar curvature [15]. More details about the PIC are provided in [5] or [20] and the references are also presented therein.…”